Math 2270-3Spring 2005Information for Exam IThis exam will consist of 120 points. The format will be T/F and computation questions. Of these computationquestions, there will be a mix of shorter and a longer problem. You will be expected to write some proo fs aswell. To study for this exam I recommend:• Going over old homework problems. I as signed a lot of problems and if you can answer all of them youshould be in good shape.• Be familiar with definitions. A good studying tip is to make flashcards of important definitions and thenreview them periodically.• I try to emphasize concepts, see how a concept relates to another and explain (at the very minimummentally) why a particular concept is important and useful. Quiz your friends and loved ones over dinnerabout concepts, or see if you can explain it to them in non-technical terms.• Reviewing proofs that we discussed in class. For example, can you wr ite out why if rref(A) = Inthen theequation Ax = b has a unique solution? See page 133 in your book-it will summarize all of the big ideaswe have covered thus far. You should be familiar with all of these statements. Notice how it says “thefollowing statements are equivalent.” This means that every statement implies every other one. Thus youshould be able to justify each statement. This interdependence is als o shown on page 109.For a more specific list of topics/motivating questions:• What are the possible types of s olutions possible for a solution set of linear equations?• How can you describe each of the types of solutions geometrically (at least in R3).• What is the technique we use to solve linear systems? How can you tell a possible type of solution setwith this method?• What is reduced row echelon fo rm?• How do you multiple matrices and vectors?• How do you add, s ubtract, and multiply matrices?• Describe the po ssible solutions for a system with (a) fewer equations than unknowns, (b) mo re equationsthan unknowns, (c) as many e quations as unknowns.• What is a linear combination?• What are linear tra ns formations? How can they be represented? What can you do to show that T (~x) isa linear tra nsformation? (Note: this last part was an important a nd instructive proof you should know.)• While I won’t ask you to k now all of the matrices of the linear transformations, I will expect you to knowhow form matrices of linear transformations from “common sense.” Meaning, what is the matrix of thelinear transformation of the projection of a vector in R2onto the x ax is? It might help to look at problems19-23 from Section 2.2.• How do you form the inverse of a linear transformation? What matrices are invertible? You should knowthe inverse of a 2 by 2 matrix.• Matrix algebra: what is the inverse of a product of invertible matrices? Are matrices commutative? (SeeSection 2.4 ).• What is the image and the kernel of a linear transformation?• Define the span of a set of vectors.• Define linear independence.• How can you tell if a set o f vectors is linearly independent?• Define basis.• How do you find a ba sis for the image and the kernel of a set of vectors?• Define dimension.• What is the rank- nullity theorem?• There won’t be any questions o n Section 3.4, however it is related to the matrix of a linear transformation.This section will appear aga in, so it is worth studying and being expose d to